Harmonic Maps and Biharmonic Maps

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.


Introduction
Harmonic maps play a central role in geometry; they are critical points of the energy functional E(ϕ) = 1  2 M |dϕ| 2 v g for smooth maps ϕ of (M, g) into (N, h).The Euler-Lagrange equations are given by the vanishing of the tension field τ (ϕ).In 1983, J. Eells and L. Lemaire extended [1] the notion of harmonic maps to biharmonic maps, which are, by definition, critical points of the bienergy functional: After G.Y. Jiang studied [2] the first and second variation formulas of E 2 , extensive studies in this area have been done (for instance, see ).Notice that harmonic maps are always biharmonic by definition.
The outline of this survey is the following: (1) Preliminaries.
(4) Harmonic maps and biharmonic maps into compact Lie groups or compact symmetric spaces.
(5) The CR analogue of harmonic maps and biharmonic maps.(6) Biharmonic hypersurfaces of compact symmetric spaces.

Preliminaries
In this section, we prepare materials for the first and second variational formulas for the bienergy functional and biharmonic maps.Let us recall the definition of a harmonic map ϕ : (M, g) → (N, h), of a compact Riemannian manifold (M, g) into another Riemannian manifold (N, h), which is an extremal of the energy functional defined by: where e(ϕ) := 1  2 |dϕ| 2 is called the energy density of ϕ.That is, for any variation {ϕ t } of ϕ with ϕ 0 = ϕ, where V ∈ Γ(ϕ −1 T N ) is a variation vector field along ϕ, which is given by V (x) = d dt | t=0 ϕ t (x) ∈ T ϕ(x) N , (x ∈ M ), and the tension field is given by τ (ϕ) = m i=1 B(ϕ)(e i , e i ) ∈ Γ(ϕ −1 T N ), where {e i } m i=1 is a locally-defined frame field on (M, g), and B(ϕ) is the second fundamental form of ϕ defined by: for all vector fields X, Y ∈ X(M ).Here, ∇ and ∇ N are connections on T M , T N of (M, g), (N, h), respectively, and ∇ and ∇ are the induced ones on ϕ −1 T N and T * M ⊗ ϕ −1 T N , respectively.By Equation (2), ϕ is harmonic if and only if τ (ϕ) = 0.
The second variation formula is given as follows.Assume that ϕ is harmonic.Then, where J is an elliptic differential operator, called the Jacobi operator acting on Γ(ϕ −1 T N ) given by: where dϕ(e i ), and R N is the curvature tensor of (N, h) for U, V ∈ X(N ).J. Eells and L. Lemaire [1] proposed polyharmonic (k-harmonic) maps, and Jiang [2] studied the first and second variation formulas of biharmonic maps.Let us consider the bienergy functional defined by: where Then, the first variation formula of the bienergy functional is given (the first variation formula) by: Here, which is called the bitension field of ϕ, and J is given in Equation (5).

Chen's Conjecture and the Generalized Chen's Conjecture
Recall the famous Chen's conjecture on biharmonic submanifold of the Euclidean space: Chen's conjecture: A biharmonic submanifold in the Euclidean space must be minimal.One can consider biharmonic submanifolds of a Riemannian manifold of non-positive curvature, and the generalized Chen's conjecture is the following (cf., R. Caddeo, S. Montaldo, P. Piu [7] and also S. Montaldo, C. Oniciuc [20], etc.): The generalized Chen's conjecture: A biharmonic submanifold in a Riemannian manifold of non-positive curvature must be minimal.
Notice that the generalized Chen's conjecture was solved negatively by giving a counter example by Ou and Tang [27,41].We first give several comments on Chen's conjecture.It should be emphasized that Chen's conjecture has been still unsolved until now.
Second, we will treat the generalized Chen's conjecture.K. Akutagawa and S. Maeta [3] gave a remarkable breakthrough to Chen's conjecture, by giving the following answer to this conjecture in the case of properly-immersed submanifolds of the Euclidean space: If we do not assume the properness condition, N. Koiso and myself [43] gave recently a partial answer in the case of generic hypersurfaces of the Euclidean space.Namely, we obtained the following: Theorem 1.Let ι : (M n , g) ⊂ E n+1 be an isometrically-immersed biharmonic hypersurface of the Euclidean space.Assume that (1) every principal curvature λ i has multiplicity one, i.e., λ i = λ j (i = j), and (2) the principal curvature vector fields For harmonic maps, it is well known that: If a domain manifold (M, g) is complete and has non-negative Ricci curvature and the sectional curvature of a target manifold (N, h) is non-positive, then every energy finite harmonic map is a constant map (cf., [44]).
See [15,[45][46][47][48][49] for recent works on harmonic maps.Therefore, it is a natural question to consider biharmonic maps into a Riemannian manifold of non-positive curvature.In this connection, Baird, Fardoun and Ouakkas (cf., [4]) showed that: If a non-compact Riemannian manifold (M, g) is complete and has non-negative Ricci curvature and (N, h) has non-positive sectional curvature, then every bienergy finite biharmonic map of (M, g) into (N, h) is harmonic.
We do not need any assumption on the Ricci curvature of (M, g) in Theorem 3. If (M, g) is a non-compact complete Riemannian manifold whose Ricci curvature is non-negative, then Vol(M, g) = ∞ (cf., Theorem 7, p. 667, [50]).Thus, Theorem 3, Equation (2), recovers the result of Baird, Fardoun and Ouakkas.Theorem 3 is sharp, since one cannot weaken the assumptions.Indeed, the generalized Chen's conjecture does not hold if (M, g) is not complete (cf., the counter examples of Ou and Tang [27]).The two assumptions of finiteness of the energy and bienergy are necessary.Indeed, there exists a biharmonic map ϕ, which is not harmonic, but the energy and bienergy are infinite.For example, is biharmonic, but not harmonic, and has infinite energy and bienergy.
As the first bi-product of our method, we obtained (cf., [21,22]): Theorem 3. Assume that (M, g) is a complete Riemannian manifold, and let ϕ : (M, g) → (N, h) be an isometric immersion; the sectional curvature of (N, h) is non-positive.If ϕ : (M, g) → (N, h) is biharmonic and M |ξ| 2 v g < ∞, then it is minimal.Here, ξ is the mean curvature normal vector field of the isometric immersion ϕ.
For the second bi-product, we can apply Theorem 3 to a horizontally-conformal submersion (cf., [51,52]).Then, we obtain: Theorem 4. Let (M m , g) be a non-compact complete Riemannian manifold (m > 2) and (N 2 , h), a Riemannian surface with non-positive curvature.Let λ be a positive function on M belonging to C ∞ (M ) ∩ L 2 (M ), and ϕ : (M, g) → (N 2 , h), a horizontally-conformal submersion with a dilation λ.If ϕ is biharmonic and λ | Ĥ| g ∈ L 2 (M ), then ϕ is a harmonic morphism.Here, Ĥ is trace of the second fundamental form of each fiber of ϕ.

Outline of the Proofs of Theorems 3-5
In this section, we first give a sketch of the proof of Theorem 3. The first step: For a fixed point x 0 ∈ M , and for every 0 < r < ∞, we first take a cut-off C ∞ function η on M satisfying that: For a biharmonic map ϕ : (M, g) → (N, h), the bitension field is given as: dϕ(e i ) = 0 (10) so we have: since the sectional curvature of (N, h) is non-positive.
The second step: Therefore, by Equation ( 11) and noticing that ∆ = ∇ * ∇, we obtain: where we used e i (η 2 ) = 2η e i (η) at the last equality.By moving the second term in the last equality of Equation ( 12) to the left-hand side, we have: where we put Now, recall the following Cauchy-Schwartz inequality: for all positive > 0 because of the inequality Therefore, for Equation ( 14), we obtain: If we put = 1 2 , we obtain, by Equations ( 13) and ( 15), Thus, by Equations ( 12) and ( 16), we obtain: The third step: Since (M, g) is complete and non-compact, we can tend r to infinity.By the assumption the right-hand side goes to zero.Furthermore, if r → ∞, the left-hand side of Equation ( 17) goes to M m i=1 |∇ e i τ (ϕ)| 2 v g since η = 1 on B r (x 0 ).Thus, we obtain: Therefore, we obtain, for every vector field X in M , Then, we have, in particular, |τ (ϕ)| is constant, say c. Because, for every vector field X on M , at each point in M , Therefore, if Vol(M, g) = ∞ and c = 0, then: which yields a contradiction.Thus, we have |τ (ϕ)| = c = 0, i.e., ϕ is harmonic.We have Equation ( 2).
Theorem 5. (Gaffney [53]) Let (M, g) be a complete Riemannian manifold.If a C 1 1-form α satisfies that M |α| v g < ∞ and M (δα) v g < ∞ or, equivalently, a C 1 vector field X defined by α Our method can be applied to an isometric immersion ϕ : (M, g) → (N, h).In this case, the one-form α defined by Equation (22) in the proof of Theorem 3 vanishes automatically without using Gaffney's theorem, since τ (ϕ) = m ξ belongs to the normal component of T ϕ(x) N (x ∈ M ), where ξ is the mean curvature normal vector field and m = dim(M ).Thus, Equation ( 24) turns out as: which implies that τ (ϕ) = m ξ = 0, i.e., ϕ is minimal.Thus, we obtain Theorem 4.
We also apply Theorem 3 to a horizontally-conformal submersion ϕ : (M m , g) → (N n , h) (m > n ≥ 2) (cf., [52,54]).In the case that a Riemannian submersion from a space form of constant sectional curvature into a Riemann surface (N 2 , h), Wang and Ou (cf., [19,28]) showed that it is biharmonic if and only if it is harmonic.We treat with a submersion from a higher dimensional Riemannian manifold (M, g) (cf., [51]).Namely, let ϕ : M → N be a submersion, and each tangent space T x M (x ∈ M ) is decomposed into the orthogonal direct sum of the vertical space V x = Ker(dϕ x ) and the horizontal space H x : and we assume that there exists a positive C ∞ function λ on M , called the dilation, such that, for each The map ϕ is said to be horizontally homothetic if the dilation λ is constant along horizontally curves in M .
If ϕ : (M m , g) → (N n , h) (m > n ≥ 2) is a horizontally-conformal submersion, then, the tension field τ (ϕ) is given (cf., [51,52]) by: where grad H 1 λ 2 is the H-component of the decomposition according to Equation ( 28) of grad 1 and Ĥ is the trace of the second fundamental form of each fiber, which is given by Ĥ = , where a local orthonormal frame field {e i } m i=1 on M is taken in such a way that {e ix |i = 1, • • • , n} belong to H x and {e j x |j = n + 1, • • • , m} belong to V x where x is in a neighborhood in M .Then, due to Theorems 3 and Equation ( 29), we have immediately: Theorem 6.Let (M m , g) be a complete non-compact Riemannian manifold and (N n , h) a Riemannian manifold with the non-positive sectional curvature (m > n ≥ 2).Let ϕ : (M, g) → (N, h) be a horizontally-conformal submersion with the dilation λ satisfying that: Assume that, either M λ 2 v g < ∞ or Vol(M, g) = M v g = ∞.Then, if ϕ : (M, g) → (N, h) is biharmonic, then it is a harmonic morphism.
Due to Theorem 7, we have: Corollary 7. Let (M m , g) be a complete non-compact Riemannian manifold and (N 2 , h) a Riemannian surface with the non-positive sectional curvature (m > n = 2).Let ϕ : (M, g) → (N, h) be a horizontally-conformal submersion with the dilation λ satisfying that: Then, if ϕ : (M, g) → (N, h) is biharmonic, then it is a harmonic morphism.
(2) For a biharmonic map of (M, g) into (N, h), the non-positivity of (N, h) implies that: which is stronger than the Bochner-type formula |τ (ϕ)| ∆|τ (ϕ)| ≥ 0. However, we can prove Theorem 3 in an alternative way by using the latter one.Here, ∆ = m i=1 (e i 2 −∇ e i e i ) denotes the negative Laplace operator acting on C ∞ (M ).

Harmonic Maps and Biharmonic Maps into Compact Lie Groups or Symmetric Spaces
In this section, we treat with harmonic maps and biharmonic maps into compact Lie groups or symmetric spaces of the compact type.

Biharmonic Maps into Compact Lie Groups
We first treat with harmonic maps and biharmonic maps into compact Lie groups.Let θ be the Maurer-Cartan form on G, i.e., a g-valued left invariant one-form on G, which is defined by θ y (Z y ) = Z, (y ∈ G, Z ∈ g).For every C ∞ map ψ of (M, g) into (G, h), let us consider a g-valued one-form α on M given by α = ψ * θ.Then, it is well known (see for example, [55]) that: where α = ψ * θ and θ is the Maurer-Cartan form of G. Thus, ψ : (M, g) → (G, h) is harmonic if and only if δα = 0.
Furthermore, let {X s } n s=1 be an orthonormal basis of g with respect to the inner product , .Then, for every for all x ∈ M .Then, for every X ∈ X(M ), where we regarded a vector field Y ∈ X(G) by Y (x) = Y (ψ(x)) (x ∈ M ) to be an element in the space Here, let us recall that the Levi-Civita connection ∇ h of (G, h) is given (cf., [56,57] Volume II, p. 201, Theorem 3.3) by: where the structure constant C ts of g is defined by [X t , X s ] = n =1 C ts X , and satisfies that: Thus, we have by Equations ( 37) and (38), because we have: and Substituting Equations ( 40) and ( 41) into the above, we have Equation (39).Therefore, we obtain the following together with Equations ( 36) and (39).

Biharmonic Maps into Compact Symmetric Spaces
Now, let θ be the Maurer-Cartan form on G, i.e., a g-valued left invariant one-form on G, which is defined by θ y (Z y ) = Z (y ∈ G, Z ∈ g).For every C ∞ map ϕ of (M, g) into (G/K, h) with a lift ψ : M → G, let us consider a g-valued one-form α on M given by α = ψ * θ and the decomposition: corresponding to the decomposition g = k ⊕ m.Then, it is well known (see, for example, [55]) that: where α = ϕ * θ, θ is the Maurer-Cartan form of G and δ(α m ) is the co-differentiation of the m-valued one-form α m on (M, g).Thus, ϕ : (M, g) → (G/K, h) is harmonic if and only if: Furthermore, we obtain: Theorem 14.We have: where ∆ g is the (positive) Laplacian of (M, g) acting on C ∞ functions on M , and {e i } m i=1 is a local orthonormal frame field on (M, g).
Therefore, we obtain immediately the following two corollaries.
Here, T is the characteristic vector field of (M, g θ ), Several examples of pseudo-biharmonic immersions of (M, g θ ) into the unit sphere or complex projective space are given in [31].

Explanations of Notions and Proofs of the CR Rigidity
We explain the terminologies in the above results following Dragomir and Montaldo [10] and also Barletta, Dragomir and Urakawa [60].We also prepare the materials on pseudo-harmonic maps and pseudo-biharmonic maps (see also [61]).
Let M be a strictly pseudoconvex CR manifold of (2n + 1)-dimension, T the characteristic vector field on M , J the complex structure of the subspace H x (M ) of T x (M ) (x ∈ M ) and g θ the Webster-Riemannian metric on M defined for X, Y ∈ H(M ) by: Let us recall for a C ∞ map ϕ of (M, g θ ) into another Riemannian manifold (N, h); the pseudo-energy E b (ϕ) is defined [60] by: where {X i } 2n i=1 is an orthonormal frame field on (H(M ), g θ ).Then, the first variational formula of E b (ϕ) is as follows [60].For every variation {ϕ t } of ϕ with ϕ 0 = ϕ, where Here, τ b (ϕ) is the pseudo-tension field, which is given by: where The second variational formula of E b is given as follows ([60], p.733): where J b is a subelliptic operator acting on Γ(ϕ −1 T N ) given by: Here, for V ∈ Γ(ϕ −1 T N )), where ∇ is the Tanaka-Webster connection, ∇ the induced connection on φ −1 T N induced from the Levi-Civita connection ∇ h and {X i } 2n i=1 a local orthonormal frame field on (H(M ), g θ ), respectively.Here, (∇ Dragomir and Montaldo [10] introduced the pseudo-bienergy given by: where τ b (ϕ) is the pseudo-tension field of ϕ.They gave the first variational formula of E b,2 as follows ([10], p. 227): where τ b,2 (ϕ) is called the pseudo-bitension field given by: Then, a smooth map ϕ of (M, g θ ) into (N, h) is said to be pseudo-biharmonic if τ b,2 (ϕ) = 0.By definition, a pseudo-harmonic map is always pseudo-biharmonic.Theorem 20. (cf., Theorem 18) Assume that ϕ is a pseudo-biharmonic map of a strictly pseudoconvex complete CR manifold (M, g θ ) into another Riemannian manifold (N, h) of non-positive curvature.
(Proof of Theorem 21) The proof is divided into several steps.The first step: For an arbitrarily fixed point x 0 ∈ M , let B r (x 0 ) = {x ∈ M : r(x) < r} where r(x) is a distance function on (M, g θ ), and let us take a cut-off function η on (M, g θ ), i.e., where r is the distance function and ∇ g θ is the Levi-Civita connection of (M, g θ ), respectively.Assume that ϕ : (M, g θ ) → (N, h) is a pseudo-biharmonic map, i.e., Here, let us recall, for V, W ∈ Γ(ϕ −1 T N )), where {e α } 2n+1 α=1 is a locally-defined orthonormal frame field of (M, g θ ), X j 2n j=1 is an orthonormal frame of H(M ) and ∇ H X W (X ∈ X(M ), W ∈ Γ(ϕ −1 T N )) is defined by: Here, X H is the H(M )-component of X corresponding to the decomposition of T x (M ) = H x (M ) ⊕ RT x (x ∈ M ), and ∇ is the induced connection of ϕ −1 T N from the Levi-Civita connection ∇ h of (N, h).Since where we define V j , W j ∈ Γ(ϕ −1 T N ) (j = 1, • • • , 2n) by: Then, since it holds that 0 ≤ √ V i ± 1 √ W i 2 for every > 0, we have, RHS of Equation ( 72 g θ (X i , X j ) h(dϕ(X i ), dϕ(X j )) = 2n i=1 h(dϕ(X i ), dϕ(X i )) and